Proving the Tangent Addition Formula: tan(u v)

To prove the tangent addition formula which states that:

(tan(u v) frac{tan u tan v}{1 - tan u tan v})

we will start from the definitions of sine and cosine functions. The tangent function is defined as the ratio of sine to cosine:

(tan x frac{sin x}{cos x})

Step-by-Step Proof

Start with the definition of tangent:

(tan(u v) frac{sin(u v)}{cos(u v)})

Using the Sine and Cosine Addition Formulas

The sine addition formula and cosine addition formulas are:

(sin(u v) sin u cos v cos u sin v)

(cos(u v) cos u cos v - sin u sin v)

Substitute these formulas into the tangent expression:

(tan(u v) frac{sin u cos v cos u sin v}{cos u cos v - sin u sin v})

Dividing the Numerator and Denominator by (cos u cos v)

Divide the numerator and denominator by (cos u cos v):

(tan(u v) frac{frac{sin u}{cos u} frac{sin v}{cos v}}{1 - frac{sin u}{cos u} cdot frac{sin v}{cos v}})

Recognizing (frac{sin u}{cos u} tan u) and (frac{sin v}{cos v} tan v)

Recognize that:

(frac{sin u}{cos u} tan u) and (frac{sin v}{cos v} tan v)

This simplifies the expression to:

(tan(u v) frac{tan u tan v}{1 - tan u tan v})

Conclusion

Thus, we have proved that:

(tan(u v) frac{tan u tan v}{1 - tan u tan v})

This formula is useful in various applications in trigonometry, calculus, and physics.

Extending the Proof to Other Trigonometric Functions

Since (tan theta frac{sin theta}{cos theta}), we must first find and prove the angle sum identities of (sin(a b)) and (cos(a b)).

Using the Fact that (e^{itheta} cos theta i sin theta)

We have (e^{ia} e^{ib} e^{iab}), which means:

((cos a i sin a)(cos b i sin b) cos(ab) i sin(ab))

Expanding the left side yields:

(cos a sin b i sin a cos b i cos a sin b - sin a sin b cos(ab) i sin(ab))

Since the real part on the left hand side must equal the real part on the right and the imaginary parts on the left must equal the imaginary parts on the right, we have:

(sin(a b) sin a cos b cos a sin b)

(cos(a b) cos a cos b - sin a sin b)

Now Finding and Proving the Tangent Addition Formula

(tan(a b) frac{sin(a b)}{cos(a b)})

(tan(a b) frac{sin a cos b cos a sin b}{cos a cos b - sin a sin b})

Dividing Both Numerator and Denominator by (cos a cos b)

Dividing both numerator and denominator on the right hand side by (cos a cos b) yields:

(tan(a b) frac{frac{sin a}{cos a} frac{sin b}{cos b}}{1 - frac{sin a}{cos a} cdot frac{sin b}{cos b}})

This Means that (tan(a b) frac{tan a tan b}{1 - tan a tan b})

ADITIONAL:

One can use the fact that (sin(-theta) -sin(theta)) and (cos(-theta) cos(theta)) to derive the angle subtraction identity for (sin(a - b)) and (cos(a - b)) and subsequently for (tan(a - b)).